Endovascular navigation is a crucial aspect of minimally invasive procedures, where precise control of curvilinear instruments like guidewires is critical for successful interventions. A key challenge in this task is accurately predicting the evolving shape of the guidewire as it navigates through the vasculature, which presents complex deformations due to interactions with the vessel walls. Traditional segmentation methods often fail to provide accurate real-time shape predictions, limiting their effectiveness in highly dynamic environments. To address this, we propose SplineFormer, a new transformer-based architecture, designed specifically to predict the continuous, smooth shape of the guidewire in an explainable way. By leveraging the transformer's ability, our network effectively captures the intricate bending and twisting of the guidewire, representing it as a spline for greater accuracy and smoothness. We integrate our SplineFormer into an end-to-end robot navigation system by leveraging the condensed information. The experimental results demonstrate that our SplineFormer is able to perform endovascular navigation autonomously and achieves a 50% success rate when cannulating the brachiocephalic artery on the real robot.